In this thesis, we propose that a ferro-orbital order, which breaks the degeneracy between the Fe $d_{xz}$ and $d_{yz}$ orbitals, is the effective cause of the structural and the magnetic transitions in the iron-based superconductors. We will discuss this orbital order in the framework of the local-itinerant dichotomy. First, due to the spatial anisotropy of the occupied orbitals that form the local moments, the magnetic exchange constants acquire dramatically different values along the two in-plane directions. Second, the itinerant electrons also undergo a nematic transition, causing the anisotropy observed in various experiments. Finally, combining orbital order in both the local moments and itinerant electrons, we find that the underlying magnetism is unfrustrated, consistent with the inelastic neutron scattering results.
The thesis is organized as follows. We will first provide the necessary background knowledge of the iron-based superconductors in Chapter 1. As a preliminary, we discuss in detail three different theoretical approaches, namely the weak-coupling, strong-coupling and local-itinerant models. Chapter 2 serves as the motivation of the thesis. Various experimental results will be presented to demonstrate the existence of the in-plane anisotropy. We will introduce two distinct theoretical scenarios that account for the nematic order. We will argue that orbital order, instead of the spin-nematic order, is the underlying mechanism. Chapters 3, 4, and 5 are the main contents of the thesis. In Chapter 3, we will study the orbital order from the strong-coupling theories, with emphasis on the Kugel-Khomskii model. Chapter 4 deals with the orbital order in the weak-coupling theories and its experimental consequences. Finally in Chapter 5, we propose the degenerate double-exchange model, and show how the orbital order in the itinerant electrons leads to the unfrustrated effective spin model.